= Good-prime and bad-prime Hecke coefficient formula
{title2=$a_n(T_pf)=a_{pn}+\chi(p)p^{k-1}a_{n/p}$}
At a prime not dividing the level, the root-of-unity average contributes $a_{pn}$ and the scaling overlattice contributes the second term. At a prime dividing the level that overlattice loses the marked point's exact order and is excluded. Extending the <Dirichlet character> by zero gives a uniform formula, with $a_{n/p}=0$ unless $p\mid n$. The bad-prime operator is $U_p$.
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